Ising model on trees and factors of IID

We study the ferromagnetic Ising model on the infinite $d$-regular tree under the free boundary condition. This model is known to be a factor of IID in the uniqueness regime, when the inverse temperature $β\ge 0$ satisfies $\tanh β\le (d-1)^{-1}$. However, in the reconstruction regime ($\tanh β> (d-1)^{-\frac{1}{2}}$), it is not a factor of IID. We construct a factor of IID for the Ising model beyond the uniqueness regime via a strong solution to an infinite dimensional stochastic differential equation which partially answers a question of Lyons. The solution $\{X_t(v) \}$ of the SDE is distributed as $X_t(v) = tτ_v + B_t(v),$ where $\{τ_v \}$ is an Ising sample and $\{B_t(v) \}$ are independent Brownian motions indexed by the vertices in the tree. Our construction holds whenever $\tanh β\le c(d-1)^{-\frac{1}{2}}$, where $c>0$ is an absolute constant.

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